Chapter 11: Zero, the Discontinuity

This means there is no tolerance for zero, right?

This means there is no tolerance for zero, right?

Zero is not a number. Oh, sure, it looks like a number, and we use it like a number, but zero is not a number.

Zero has two distinct functions in our mathematical language system. The first, and most logical, as was mentioned in the chapter about Number Theory, is as a place holder. Zero, when used in notation means ‘none of these’. For example, if the number is 100, then the zeros in it mean: no tens, and no ones. Alternately if we choose the number 0.01, then the zeros mean ‘no ones and no tenths’. This is very useful, and it is a sound and powerful notational concept.

But the other meaning for zero is the one that causes all of the problems. Zero also means nothing, and when it is used to mean nothing it causes all sorts of trouble. In this sense, it means ‘no value at all’, and this is what makes it so subversive.

Looking back at the example of temperature measurement, let’s examine how zero relates to the system. All known matter in the physical world has some positive value temperature, because temperature is one of the primary measurements that we utilize to gauge the energy that atoms have, or more succinctly, atomic motion. Atoms, and the subatomic particles that comprise them are, in all temperatures above zero, constantly in motion. Collectively, they vibrate. Viewed individually, they move. An electron, though not orbiting the nucleus of an atom in the classical sense, is constantly changing position within its prescribed field. All of these motions together are summed up in our concept of temperature.

It’s a little hard to grasp, but the reason that a black metal park bench will burn your posterior on a hot sunny day is that the light from the sun has transmitted energy to the bench, which in turn has caused the atoms from which the bench is made to move more quickly within the particular lattice that restrains them. We feel this increase in energy, in motion, as heat, and can detect small changes in the overall energy of the object through the nerves in our skin as ‘warmth’. My coffee cup is exciting the atoms of my table as I write this. Add enough of this energy to the object, and it will change state. At some point, the park bench will melt (although not by the energy from the sun, unless concentrated). The table will start to pyrolize, that is, to off gas hydrocarbons (smoke). And if the bench were hot enough, it would overwhelm your body’s defenses and change the state of the compounds that compose your skin. We call this cooking. So temperature, or warmth, is a measurement of the collective ‘rest’ energy (nice oxymoron) of any ‘thing’ that you want to measure, from electrons to cosmic clouds of interstellar dust.

As pointed out before, this measurement has no practical upper limit. We surmise that the matter at the edge of a black hole is raised to billions of degrees of temperature. So too, with the particles that we smash in cyclotrons. But the lower limit is defined as the point where all of the motion stops, and we call this, Absolute Zero.

Absolute Zero is a point that is easily defined, but never achieved. It is a concept, an idea, a theoretical point at which all ‘things’ stop moving. But can we ever actually get there? Currently, the answer is: no. True, using clever tools and techniques, we can get to within a few millionths of a degree of Absolute Zero. There is much valuable research regarding the fundamental properties of matter that is being conducted at this energy level but, as a practical matter, Absolute Zero is an asymptotic point, a point that is always just another fraction of a degree away. The search for it is a quest that knows no end.

And so it is with zero the ‘number’, because in the quest for nothingness, even a little bit of something will keep you from ever getting there, and fundamentally speaking, if there is matter, and matter equals energy (thanks, Dr. Einstein), then something made of matter can never be cooled to Absolute Zero, because there will still be energy, hence temperature.

Therefore, zero is not a ‘place’ or a value that can ever be known or measured. It is a concept, and not, like all the other values that we use, a number.

Of course, one example does not ordinarily prove a case, and so it should be with zero. But other examples abound. Take velocity for instance. In this case, we’re referring to macroscopic velocity, the kind that you can see rather than feel. (As we discussed earlier, there is some question as to whether or not humans can actually see velocity, but for the time being, let’s just assume that we can.) Is there any such thing as a zero velocity?

Actually, this topic has been at the heart of a subject for debate for millennia and relates directly to the static earth concept. But, if one looks at this from a quasi-cosmological perspective, you have to say that the answer is, again, no.

Everything is in motion.

This was discussed briefly in the first chapter on geometry, but it is worth repeating here. The universe, as we know it, is flying apart. All of the components of the perceivable universe are moving away from each other at incredible speeds. There are some notable exceptions: galaxies that are moving toward a collision with each other, binary stars in their death dances, comets that hammer planets. But for the most part, the universe is getting larger and everything in it is getting farther apart, like dots on the surface of an inflating balloon. Every galaxy has its own velocity that appears to be in a direction that is away from a postulated center. And therein lays the rub. Since every galaxy is moving, then there can be no zero velocity within it, at least from an absolute perspective.

And within each and every galaxy, all of its constituents are moving, too. Most of them rotate within the structure, but even those that don’t are in a constant state of flux, as gravity (and probably the electric force) cause them to shift and adjust within the confines of their groups. And as you go down the line to the stars and their planetary systems, and to the planets themselves, all you see is movement.

There just is no, can be no, velocity that is truly zero.

Of course, within the confines of a smaller system, like within a room, or within a small segment of a planet, there is such a thing as a relative velocity that equals zero; but here we are talking about conceptual nothingness; the concept as to whether or not there can be a rest velocity that truly equals zero, and the answer is equivocally, no.

So we have this concept of nothingness that we call zero, that is, as a practical matter, unachievable, and we’ve located it exactly where in our current number system? Why, right smack dab in the middle, of course! Geometrically speaking, zero is at the center of our whole notational system (an interesting philosophical and epistemological concept), yet by any measure, it is undefined.

But it’s worse than that; because as we discussed in the chapter on number theory, division by zero has been defined an illegal operation, since there is no definition for the outcome of it. If you plot the reciprocal of all of the numbers on a number line, it becomes quite clear, the graph becomes asymptotic at zero. Then, as you pass zero and travel into the (evil) negative realm, somehow our answers return back from infinity, and are once again, well behaved.

Engineers and mathematicians call this sort of thing a discontinuity, and that is what we’ve been driving at here. Zero, being a concept that lacks a clear chartable value, cannot logically be at the center of any geometric or numerical system. Zero is an end point. It is no better defined in reality than the concept of ‘infinity’, and to utilize it as the very foundation of our notational systems is not only wrong, it is misleading and dangerous, because it leads to an unsustainable train of logic that obscures the true nature of the environment and the universe that surrounds us.

What is needed is a scientific notational and measurement system that does not need to be anchored to an indefinable concept. A system that is based and focused on measurements that are knowable and well defined. A system that does not require second order calculations to determine first order answers, and one that truly codifies and accounts for all of the measurements needed to find the required answers. Such a system does exist, and we will outline it in future chapters. But first, we need to discuss some of the aspects of Relativity and Quantum Mechanics, as they have relevance to the whole, and the best way to start is to go back to the thoughts of the fabulous Dr. Einstein.

Whew. We’ve finally finished the sections that have been review for most of you, although I’m not sure that many of you realized the problems with and the implications of the concept of zero. I know that I didn’t until I started working on this geometry.

Zero is, by definition, unknowable.

But enough about that. In the next sections we’re going to take a very subjective look at Special Relativity, the Photoelectric Effect and take a little romp (a very little romp) through Quantum Theory. These topics are difficult to understand fully, but having some knowledge of these realms of thought is important to our overall goals and the reasons that the new perspective presented here is more accurate and elegant than the geometry that we commonly use.

One other note: try not to get too bogged down in the mathematics presented here. I’ve put in what I thought was just enough to let the science nerds among us know that I understand the language and can utilize the methodology, but the formulas are not that important to the text, well, at least until we get to the chapter on Coriolis anyway.

Thanks for making it this far. The tale gets more interesting as we move on from here.

– O. Penurmind

Up next: Chapter 12: Special Relativity

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